Weighted Pivot Coordinates for Compositional Data and Their Application to Geochemical Mapping


The log ratio methodology converts compositional data, such as concentrations of chemical elements in a rock, from their original Aitchison geometry to interpretable real orthonormal coordinates, thereby allowing meaningful statistical processing and visualization. However, it must be taken into account that the original concentrations can be flawed by detection limit or imprecision problems that can severely affect the resulting coordinates. This paper aims to construct such orthonormal log ratio coordinates, called weighted pivot coordinates, that capture the relevant relative information about an original component and treat the redundant information in a controlled manner. Theoretical developments are supported by a thorough simulation study. Weighted pivot coordinates are then applied to the geochemical mapping of catchment outlet sediments from the National Geochemical Survey of Australia illustrating their advantage over possible alternatives.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. Aitchison J (1986) The statistical analysis of compositional data. Monographs on statistics and applied probability. Chapman and Hall, London

    Google Scholar 

  2. Blake D, Kilgour B (1998) Geological regions of Australia 1:5,000,000 scale (dataset). Geoscience Australia, Canberra. http://www.ga.gov.au/metadata-gateway/metadata/record/gcat_a05f7892-b237-7506-e044-00144fdd4fa6/Geological+Regions+of+Australia%2C+1%3A5+000+000+scale

  3. Caritat P de, Cooper M (2011a) National Geochemical Survey of Australia: the geochemical Atlas of Australia. Geoscience Australia Record, 2011/20 (2 volumes)

  4. Caritat P de, Cooper M (2011b) National Geochemical Survey of Australia: data quality assessment. Geoscience Australia Record, 2011/21 (2 volumes)

  5. Caritat P de, Grunsky EC (2013) Defining element associations and inferring geological processes from total element concentrations in Australian catchment outlet sediments: Multivariate analysis of continental-scale geochemical data. Appl Geochem 33:104126

  6. Eaton ML (1983) Multivariate statistics. A vector space approach. Wiley, New York

    Google Scholar 

  7. Egozcue JJ (2009) Reply to “On the Harker Variation Diagrams;...” by J.A. Cortés. Math Geosci 41:829–834

  8. Egozcue JJ, Pawlowsky-Glahn V (2005) Groups of parts and their balances in compositional data analysis. Math Geol 37:795–828

    Article  Google Scholar 

  9. Egozcue JJ, Pawlowsky-Glahn V (2006) Simplicial geometry for compositional data. In: Buccianti A, Mateu-Figueras G, Pawlowsky-Glahn V (eds) Compositional data analysis in the geosciences: from theory to practice, Special Publications 264. Geological Society, London, pp 145–160

  10. Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal C (2003) Isometric logratio transformations for compositional data analysis. Math Geol 35:279–300

    Article  Google Scholar 

  11. Filzmoser P, Hron K (2013) Robustness for compositional data. In: Becker C, Fried R, Kuhnt S (eds) Robustness and complex data structures. Springer, Heidelberg

    Google Scholar 

  12. Filzmoser P, Hron K (2015) Robust coordinates for compositional data using weighted balances. In: Nordhausen K, Taskinen S (eds) Modern nonparametric, robust and multivariate methods. Springer, Heidelberg, pp 167–184

  13. Filzmoser P, Hron K, Reimann C (2012) Interpretation of multivariate outliers for compositional data. Comput Geosci 39:77–85

    Article  Google Scholar 

  14. Fišerová E, Hron K (2011) On interpretation of orthonormal coordinates for compositional data. Math Geosci 43:455–468

    Article  Google Scholar 

  15. Guerreiro C, Cachão M, Pawlowsky-Glahn V, Oliveira A, Rodrigues A (2015) Compositional data analysis (CoDA) as a tool to study the (paleo)ecology of coccolithophores from coastal-neritic settings off central Portugal. Sediment Geol 319:134–146

    Article  Google Scholar 

  16. Hron K, Filzmoser P, Thompson K (2012) Linear regression with compositional explanatory variables. J Appl Stat 39:1115–1128

    Article  Google Scholar 

  17. Martín-Fernández JA, Hron K, Templ M, Filzmoser P, Palarea-Albaladejo J (2012) Model-based replacement of rounded zeros in compositional data: classical and robust approaches. Comput Stat Data Anal 56:2688–2704

    Article  Google Scholar 

  18. McKinley J, Lloyd CD (2011) Multivariate geochemical data analysis in physical geography. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester, pp 145–160

    Google Scholar 

  19. Pawlowsky-Glahn V, Buccianti A (eds) (2011) Compositional data analysis: theory and applications. Wiley, Chichester

    Google Scholar 

  20. Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015) Modeling and analysis of compositional data. Wiley, Chichester

    Google Scholar 

  21. Raymond OL (2012) Surface Geology of Australia data package 2012 edition [dataset]. Geoscience Australia, Canberra. https://www.ga.gov.au/products/servlet/controller?event=GEOCAT_DETAILS&catno=74855

  22. Reimann C, Äyräs M, Chekushin Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T (1998) Environmental Geochemical Atlas of the Central Barents Region, Geological Survey of Norway (NGU), Geological Survey of Finland (GTK), and Central Kola Expedition (CKE). Special Publication, Trondheim, Espoo, Monchegorsk

  23. Reimann C, Filzmoser P, Fabian K, Hron K, Birke M, Demetriades A, Dinelli E, Ladenberger A, GEMAS Project Team (2012) The concept of compositional data analysis in practice: total major element concentrations in agricultural and grazing land soils in Europe. Sci Total Environ 426:196–210

Download references


The insightful and constructive reviewer reports from Dr. Raimon Tolosana-Delgado and two anonymous reviewers are gratefully acknowledged; they helped to greatly improve the paper. The authors thank and acknowledge also the participants of the first GeoMap Workshop (held in Olomouc, Czech Republic, 17–20 June 2014) which provoked discussions leading to the present contribution. The paper was supported by the Grant COST Action CRoNoS IC1408 and the infrastructural part was supported from NPU I (LO1304).

Author information



Corresponding author

Correspondence to Karel Hron.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hron, K., Filzmoser, P., de Caritat, P. et al. Weighted Pivot Coordinates for Compositional Data and Their Application to Geochemical Mapping. Math Geosci 49, 797–814 (2017). https://doi.org/10.1007/s11004-017-9684-z

Download citation


  • Aitchison geometry
  • Orthonormal coordinates
  • Pivot coordinates
  • Geochemical mapping
  • National Geochemical Survey of Australia